If `P_(1)andP_(2)` be the lenghts of the perpendiculars from the origin upon the lines `xsintheta+ycostheta=(a)/(2)sin2thetaandxcostheta-ysintheta=acos2theta`, prove that `4p_(1)^(2)+p_(2)^(2)=a^(2)`.

If p and q are the lengths of perpendiculars from the origin to the lines xcostheta-ysintheta=kcos2theta and xsec""theta+y" cosec"""theta=""k , respectively, prove that p^2+4q^2=k^2 .

If p and q are the lengths of perpendiculars from the origin to the lines x cos theta - y sin theta = k cos 2 theta " and " x sec theta + y cosec theta = k , respectively, prove that p^(2) + 4q^(2) = k^(2) .

If p_(1)andp_(2) be the lenghts of the perpendiculars from the origin upon the lines 4x+3y=5cosalphaand 6x-8y=5sinalpha repectively , show that p_(1)^(2)+4p_(2)^(2)=1 .

if P is the length of perpendicular from origin to the line x/a+y/b=1 then prove that 1/(a^2)+1/(b^2)=1/(p^2)

Prove that, (1 + sin 2 theta - cos 2 theta)/(sin theta + cos theta)= 2 sin theta

If p_(1),p_(2) be the lenghts of perpendiculars from origin on the tangent and the curve x^((2)/(3))+y^((2)/(3))=a^((2)/(3)) drawn at any point on it, show that, 4p_(1)^(2)+p_(2)^(2)=a^(2)

If 2ycostheta=xsintheta and 2xsectheta-ycosectheta=3 show that x^2+4y^2=?

If p and q are the lengths of the perpendiculars from the origin to the straight lines x "sec" alpha + y " cosec" alpha = a " and " x "cos" alpha-y " sin" alpha = a "cos" 2alpha, " then prove that 4p^(2) + q^(2) = a^(2).

Show that the sum of the squares of the perpendiculars from the origin upon the lines xcosalpha+ysinalpha=acos2alphaandxsecalpha+ycossecalpha=2a is independent of alpha .

If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that 1/p^(2) = 1/a^(2) + 1/b^(2) .